Wash, pit, and dice 6-8 ripe peaches. In a medium saucepan, combine diced peaches, 1/2 cup granulated sugar, 1/2 cup water, and 1 tablespoon fresh lemon juice. Simmer over medium-low heat for 10-15 minutes until peaches are very soft.

Remove from heat, mash peaches thoroughly, then strain the mixture through a fine-mesh sieve into a large bowl to extract the smooth peach puree. Discard solids. Allow the peach puree to cool completely to room temperature.

In a small saucepan, combine 1 cup granulated sugar and 1 cup water. Heat over medium heat, stirring until sugar is completely dissolved. Remove from heat and cool completely.

Juice 8-10 medium lemons to yield about 1 1/2 cups fresh lemon juice; strain to remove any seeds or pulp. In a large pitcher, combine the cooled simple syrup with the strained lemon juice. Stir well.

Add the cooled peach puree to the lemonade base in the pitcher. Stir vigorously to combine. Begin by adding 4 cups of cold water, stir, and taste. Gradually add more cold water (up to 6 cups total) until your desired strength and sweetness are reached.

Refrigerate the peach lemonade for at least 30 minutes to thoroughly chill. Serve over plenty of ice, garnished with fresh peach slices, lemon wheels, and mint sprigs. For a sparkling version, replace some or all of the dilution water with sparkling water or club soda.